What is the general solution for cos(θ)+2cos(θ+240)=1 ?

The general solution for cos(θ)+2cos(θ+240)=1 is θ=0.61547…+360n,θ=180-0.61547…+360n

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cos(θ)+2cos(θ+240)=1

Solution

cos (θ) + 2 cos (θ + 24 0^{\circ}) = 1

Solution

θ = 0.61547 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} - 0.61547 \dots + 36 0^{\circ} n

Radians

θ = 0.61547 \dots + 2 πn, θ = π - 0.61547 \dots + 2 πn

Solution steps

cos (θ) + 2 cos (θ + 24 0^{\circ}) = 1

Rewrite using trig identities

cos (θ) + 2 cos (θ + 24 0^{\circ}) = 1

Rewrite using trig identities

cos (θ + 24 0^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (θ) cos (24 0^{\circ}) - sin (θ) sin (24 0^{\circ})

Simplify

cos (θ) cos (24 0^{\circ}) - sin (θ) sin (24 0^{\circ}) : - \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ)

cos (θ) cos (24 0^{\circ}) - sin (θ) sin (24 0^{\circ})

cos (θ) cos (24 0^{\circ}) = - \frac{1}{2} cos (θ)

cos (θ) cos (24 0^{\circ})

cos (24 0^{\circ}) = - \frac{1}{2}

cos (24 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

cos (24 0^{\circ})

Write

cos (24 0^{\circ})

cos (18 0^{\circ} + 6 0^{\circ})

= cos (18 0^{\circ} + 6 0^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= (- 1) \frac{1}{2} - 0 \cdot \frac{3}{2}

Simplify

= - \frac{1}{2}

= (- \frac{1}{2}) cos (θ)

Remove parentheses:

(- a) = - a

= - cos (θ) \frac{1}{2}

= - \frac{1}{2} cos (θ) - sin (24 0^{\circ}) sin (θ)

sin (24 0^{\circ}) = - \frac{3}{2}

sin (24 0^{\circ})

Rewrite using trig identities:

sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

sin (24 0^{\circ})

Write

sin (24 0^{\circ})

sin (18 0^{\circ} + 6 0^{\circ})

= sin (18 0^{\circ} + 6 0^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

= sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= 0 \cdot \frac{1}{2} + (- 1) \frac{3}{2}

Simplify

= - \frac{3}{2}

= - \frac{1}{2} cos (θ) - (- \frac{3}{2} sin (θ))

Apply rule

- (- a) = a

= - cos (θ) \frac{1}{2} + sin (θ) \frac{3}{2}

= - \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ)

cos (θ) + 2 (- \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ)) = 1

Simplify

cos (θ) + 2 (- \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ)) : 3 sin (θ)

cos (θ) + 2 (- \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ))

Expand

2 (- \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ)) : - cos (θ) + 3 sin (θ)

2 (- \frac{1}{2} cos (θ) + \frac{3}{2} sin (θ))

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = - \frac{1}{2} cos (θ), c = \frac{3}{2} sin (θ)

= 2 (- \frac{1}{2} cos (θ)) + 2 \cdot \frac{3}{2} sin (θ)

Apply minus-plus rules

+ (- a) = - a

= - 2 \cdot \frac{1}{2} cos (θ) + 2 \cdot \frac{3}{2} sin (θ)

Simplify

- 2 \cdot \frac{1}{2} cos (θ) + 2 \cdot \frac{3}{2} sin (θ) : - cos (θ) + 3 sin (θ)

- 2 \cdot \frac{1}{2} cos (θ) + 2 \cdot \frac{3}{2} sin (θ)

2 \cdot \frac{1}{2} cos (θ) = cos (θ)

2 \cdot \frac{1}{2} cos (θ)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2} cos (θ)

Cancel the common factor:

2

= cos (θ) \cdot 1

Multiply:

cos (θ) \cdot 1 = cos (θ)

= cos (θ)

2 \cdot \frac{3}{2} sin (θ) = 3 sin (θ)

2 \cdot \frac{3}{2} sin (θ)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 3}{2} sin (θ)

Cancel the common factor:

2

= sin (θ) 3

= - cos (θ) + 3 sin (θ)

= - cos (θ) + 3 sin (θ)

= cos (θ) - cos (θ) + 3 sin (θ)

Add similar elements:

cos (θ) - cos (θ) = 0

= 3 sin (θ)

3 sin (θ) = 1

3 sin (θ) = 1

Subtract

1

from both sides

3 sin (θ) - 1 = 0

Move

1

to the right side

3 sin (θ) - 1 = 0

Add

1

to both sides

3 sin (θ) - 1 + 1 = 0 + 1

Simplify

3 sin (θ) = 1

3 sin (θ) = 1

Divide both sides by

3

3 sin (θ) = 1

Divide both sides by

3

\frac{3 sin ( θ )}{3} = \frac{1}{3}

Simplify

\frac{3 sin ( θ )}{3} = \frac{1}{3}

Simplify

\frac{3 sin ( θ )}{3} : sin (θ)

\frac{3 sin ( θ )}{3}

Cancel the common factor:

3

= sin (θ)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

sin (θ) = \frac{3}{3}

sin (θ) = \frac{3}{3}

sin (θ) = \frac{3}{3}

Apply trig inverse properties

sin (θ) = \frac{3}{3}

General solutions for

sin (θ) = \frac{3}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

θ = arcsin (\frac{3}{3}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{3}{3}) + 36 0^{\circ} n

θ = arcsin (\frac{3}{3}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{3}{3}) + 36 0^{\circ} n

Show solutions in decimal form

θ = 0.61547 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} - 0.61547 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ)+2cos(θ+240)=1 ?
The general solution for cos(θ)+2cos(θ+240)=1 is θ=0.61547…+360n,θ=180-0.61547…+360n